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University of California, Berkeley PLASMA
A MINI-COURSE ON THE
PRINCIPLES OF LOW-PRESSURE DISCHARGES
AND MATERIALS PROCESSING
Michael A. Lieberman
Department of Electrical Engineering and Computer Science
University of California, Berkeley, CA 94720
LiebermanMinicourse10 1
University of California, Berkeley PLASMA
OUTLINE
• Introduction
• Summary of Plasma and Discharge Fundamentals
• Global Model of Discharge Equilibrium
— Break —
• Inductive Discharges
• Reactive Neutral Balance in Discharges
• Adsorption and Desorption Kinetics
• Plasma-Assisted Etch Kinetics
LiebermanMinicourse10 2
University of California, Berkeley PLASMA
INTRODUCTION TO PLASMA DISCHARGES
AND PROCESSING
LiebermanMinicourse10 3
University of California, Berkeley PLASMA
THE NANOELECTRONICS REVOLUTION
• Transistors/chip doubling every 112–2 years since 1959
• 1,000,000-fold decrease in cost for the same performance
EQUIVALENT AUTOMOTIVE ADVANCE
• 4 million km/hr
• 1 million km/liter
• Never break down
• Throw away rather than pay parking fees
• 3 cm long × 1 cm wide
• Crash 3× a day
LiebermanMinicourse10 4
University of California, Berkeley PLASMA
EVOLUTION OF ETCHING DISCHARGES
LiebermanMinicourse10 5
University of California, Berkeley PLASMA
ISOTROPIC PLASMA ETCHING
1. Start with inert molecular gas CF4
2. Make discharge to create reactive species:CF4 −→ CF3 + F
3. Species reacts with material, yielding volatile product:Si + 4F −→ SiF4 ↑
4. Pump away product
ANISOTROPIC PLASMA ETCHING
5. Energetic ions bombard trench bottom, but not sidewalls:(a) Increase etching reaction rate at trench bottom(b) Clear passivating films from trench bottom
Mask
PlasmaIons
LiebermanMinicourse10 6
University of California, Berkeley PLASMA
PLASMA DENSITY VERSUS TEMPERATURE
LiebermanMinicourse10 7
University of California, Berkeley PLASMA
RELATIVE DENSITIES AND ENERGIES
Charged particle densities ≪ neutral particle densities
LiebermanMinicourse10 8
University of California, Berkeley PLASMA
NON-EQUILIBRIUM
• Energy coupling between electrons and heavy particles is weak
Input
Electrons Ions
Neutrals
strong
power
Walls
Walls
Walls
strong
strongweak
weakweak
• Electrons are not in thermal equilibrium with ions or neutrals
Te ≫Ti in plasma bulk
Bombarding Ei ≫ Ee at wafer surface
• “High temperature processing at low temperatures”1. Wafer can be near room temperature2. Electrons produce free radicals =⇒ chemistry3. Electrons produce electron-ion pairs =⇒ ion bombardment
LiebermanMinicourse10 9
University of California, Berkeley PLASMA
ELEMENTARY DISCHARGE BEHAVIOR
• Uniform density of electrons and ions ne and ni at time t = 0• Low mass warm electrons quickly drain to the wall, forming sheaths
• Ion bombarding energy Ei= plasma-wall potential Vp
• Separation into bulk plasma and sheaths occurs for ALL dischargesLiebermanMinicourse10 10
University of California, Berkeley PLASMA
SUMMARY OF PLASMA FUNDAMENTALS
LiebermanMinicourse10 11
University of California, Berkeley PLASMA
THERMAL EQUILIBRIUM PROPERTIES
• Electrons generally near thermal equilibriumIons generally not in thermal equilibrium
• Maxwellian distribution of electrons
fe(v) = ne
(m
2πkTe
)3/2
exp
(−mv2
2kTe
)
where v2 = v2x + v2
y + v2z
fe(vx)
vxvTe =(kTe/m)1/2
• Pressure p = nkTFor neutral gas at room temperature (300 K)
ng[cm−3] ≈ 3.3× 1016 p[Torr]
LiebermanMinicourse10 12
University of California, Berkeley PLASMA
AVERAGES OVER MAXWELLIAN DISTRIBUTION
• Average energy〈 12mv2〉 = 1
ne
∫d3v 1
2mv2fe(v) = 32kTe
• Average speed
ve =1
ne
∫d3v vfe(v) =
(8kTe
πm
)1/2
• Average electron flux lost to a wall
x
y
z Γe [m–2s–1]
Γe =
∫ ∞
−∞
dvx
∫ ∞
−∞
dvy
∫ ∞
0
dvzvzfe(v) =1
4neve [m−2-s−1]
• Average kinetic energy lost per electron lost to a wall
Ee = 2 Te
LiebermanMinicourse10 13
University of California, Berkeley PLASMA
FORCES ON PARTICLES
• For a unit volume of electrons (or ions)
mnedue
dt= qneE−∇pe −mneνmue
mass × acceleration = electric field force ++ pressure gradient force + friction (gas drag) force
• m = electron massne = electron densityue = electron flow velocityq = −e for electrons (+e for ions)E = electric fieldpe = nekTe = electron pressureνm = collision frequency of electrons with neutrals
x
pe
pe(x) pe(x + dx) ue
Dragforce
Neutrals
x x + dx
∇pe
LiebermanMinicourse10 14
University of California, Berkeley PLASMA
BOLTZMANN FACTOR FOR ELECTRONS
• If electric field and pressure gradient forces almost balance0 ≈ −eneE−∇pe
• Let E = −∇Φ and pe = nekTe
∇Φ =kTe
e
∇ne
ne• Put kTe/e = Te (volts) and integrate to obtain
ne(r) = ne0 eΦ(r)/Te
Φ
x
mx
ne
ne0
LiebermanMinicourse10 15
University of California, Berkeley PLASMA
PLASMA DIELECTRIC CONSTANT ǫp
• RF discharges are driven at a frequency ω
E(t) = Re (E ejωt), etc.
• Define ǫp from the total current in Maxwell’s equations
∇× H = Jc + jωǫ0E︸ ︷︷ ︸ ≡ jωǫpE
Total current J
• Conduction current is Jc = −eneue
Newton’s law is jωmue = −eE −mνmue
Solve for ue and evaluate Jc to obtain
ǫp ≡ ǫ0κp = ǫ0
[1− ω2
pe
ω(ω − jνm)
]
with ωpe = (e2ne/ǫ0m)1/2 the electron plasma frequency
• For ω ≫ νm, ǫp is mainly real (nearly lossless dielectric)
LiebermanMinicourse10 16
University of California, Berkeley PLASMA
PLASMA CONDUCTIVITY σp
• It is useful to introduce rf plasma conductivity Jc ≡ σpE
• Since Jc is a linear function of E [p. 16]
σp =e2ne
m(νm + jω)
• DC plasma conductivity (ω ≪ νm)
σdc =e2ne
mνm
• RF current flowing through the plasma heats electrons(just like a resistor)
LiebermanMinicourse10 17
University of California, Berkeley PLASMA
SUMMARY OF DISCHARGE FUNDAMENTALS
LiebermanMinicourse10 18
University of California, Berkeley PLASMA
ELECTRON COLLISIONS WITH ARGON
• Maxwellian electrons collide with Ar atoms (density ng)# collisions of a particular kind
s-m3 = νne = Kng ne
ν = collision frequency [s−1], K(Te) = rate coefficient [m3/s]
• Electron-Ar collision processese + Ar −→ Ar+ + 2e (ionization)e + Ar −→ e + Ar∗ −→ e + Ar + photon (excitation)e + Ar −→ e + Ar (elastic scattering) e
ArAr
e
• Rate coefficient K(Te) is average of cross section σ(vR) [m2]over Maxwellian distribution
K(Te) = 〈σ vR〉Maxwellian
vR = relative velocity of colliding particles
LiebermanMinicourse10 19
University of California, Berkeley PLASMA
ELECTRON-ARGON RATE COEFFICIENTS
LiebermanMinicourse10 20
University of California, Berkeley PLASMA
ION COLLISIONS WITH ARGON
• Argon ions collide with Ar atomsAr+ + Ar −→ Ar+ + Ar (elastic scattering)Ar+ + Ar −→ Ar + Ar+ (charge transfer)
ArAr
Ar+
Ar+
Ar
Ar
Ar+
Ar+
• Total cross section for room temperature ions σi ≈ 10−14 cm2
• Ion-neutral mean free path (distance ion travels before colliding)
λi =1
ngσi
• Practical formula
λi(cm) =1
330 p, p in Torr
LiebermanMinicourse10 21
University of California, Berkeley PLASMA
THREE ENERGY LOSS PROCESSES
1. Collisional energy Ec lost per electron-ion pair created
KizEc = KizEiz + KexEex + Kel(2m/M)(3Te/2)
=⇒ Ec(Te) (voltage units)
Eiz, Eex, and (3m/M)Te are energies lost by an electron due to anionization, excitation, and elastic scattering collision
2. Electron kinetic energy lost to walls
Ee = 2 Te
3. Ion kinetic energy lost to walls is mainly due to the dc potential Vs
across the sheathEi ≈ Vs
• Total energy lost per electron-ion pair lost to walls
ET = Ec + Ee + EiLiebermanMinicourse10 22
University of California, Berkeley PLASMA
COLLISIONAL ENERGY LOSSES
LiebermanMinicourse10 23
University of California, Berkeley PLASMA
BOHM (ION LOSS) VELOCITY uB
Plasma Sheath Wall
Density ns
uB
• Due to formation of a “presheath”, ions arrive at the plasma-sheathedge with directed energy kTe/2
1
2Mu2
i =kTe
2
• Electron-ion pairs are lost at the Bohm velocity at the plasma-sheathedge (density ns)
ui = uB =
(kTe
M
)1/2
LiebermanMinicourse10 24
University of California, Berkeley PLASMA
STEADY STATE DIFFUSION
• Particle balance: losses to walls = creation in volume
∇ · Γe,i = νizne
• Ambipolar: equal fluxes (and densities) of electrons and ions
Γ = −Da∇nDa = kTe/Mνi = ambipolar diffusion coefficient
• Boundary condition: Γwall = nsuB at plasma-sheath edge
0x
ns
n0
Γwall Γwall
l/2−l/2
=⇒ hl ≡ ns/n0 = edge-to-center density ratio
LiebermanMinicourse10 25
University of California, Berkeley PLASMA
PLASMA DIFFUSION AT LOW PRESSURES
• Plasma density profile is relatively flat in the center and fallssharply near the sheath edge
0x
ns
n0
Γwall Γwall
l/2−l/2
• Ion and electron loss flux to the wall isΓwall = nsuB ≡ hln0uB
• The edge-to-center density ratio is
hl ≡ns
n0≈ 0.86
(3 + l/2λi)1/2
where λi = ion-neutral mean free path [p. 21]• Applies for pressures < 100 mTorr in argon
LiebermanMinicourse10 26
University of California, Berkeley PLASMA
PLASMA DIFFUSION INLOW PRESSURE CYLINDRICAL DISCHARGE
R
l
Plasma
ne = ni = n0
nsl = hl n0
nsR = hR n0
• Loss fluxes to the axial and radial walls areΓaxial = hln0uB , Γradial = hRn0uB
where the edge-to-center density ratios are
hl ≈0.86
(3 + l/2λi)1/2
, hR ≈0.8
(4 + R/λi)1/2
• Applies for pressures < 100 mTorr in argon
LiebermanMinicourse10 27
University of California, Berkeley PLASMA
GLOBAL MODEL OF DISCHARGE EQUILIBRIUM
LiebermanMinicourse10 28
University of California, Berkeley PLASMA
PARTICLE BALANCE AND Te
• Assume uniform cylindrical plasma absorbing power Pabs
R
l
PlasmaPabs
ne = ni = n0
• Particle balance
Production due to ionization = loss to the walls
Kizngn0πR2l = (2πR2hln0 + 2πRlhRn0)uB/ / /
• Solve to obtain
Kiz(Te)
uB(Te)=
1
ngdeff
where
deff =1
2
Rl
Rhl + lhR
is an effective plasma size• Given ng and deff =⇒ electron temperature Te
• Te varies over a narrow range of 2–5 voltsLiebermanMinicourse10 29
University of California, Berkeley PLASMA
ELECTRON TEMPERATURE IN ARGON DISCHARGE
LiebermanMinicourse10 30
University of California, Berkeley PLASMA
ION ENERGY FOR LOW VOLTAGE SHEATHS
• Ei = energy entering sheath + energy gained traversing sheath• Ion energy entering sheath = Te/2 (voltage units)• Sheath voltage determined from particle conservation
Plasma Sheath
+ −
Γi Γi
Γe
Vs
Insulatingwall
Density ns~ 0.2 mm
Γi = nsuB , Γe = 14nsve︸ ︷︷ ︸
e−V s/Te
with ve = (8eTe/πm)1/2 Random fluxat sheath edge
• The ion and electron fluxes at the wall must balance
V s =Te
2ln
(M
2πm
)
or V s ≈ 4.7 Te for argon• Accounting for the initial ion energy, Ei ≈ 5.2 Te
LiebermanMinicourse10 31
University of California, Berkeley PLASMA
ION ENERGY FOR HIGH VOLTAGE SHEATHS
• Large ion bombarding energies can be gained nearrf-driven electrodes
+– –+
Clarge
+–
~Vrf
~Vrf
Low voltagesheath ~ 5.2 Te
Plasma
Plasma
s
Vs
VsVs
Vs ~ 0.4 Vrf
Vs ~ 0.8 Vrf
• The sheath thickness s is given by the Child law
Ji = ensuB =4
9ǫ0
(2e
M
)1/2V
3/2s
s2
• Estimating ion energy is not simple as it depends on the type ofdischarge and the application of rf or dc bias voltages
LiebermanMinicourse10 32
University of California, Berkeley PLASMA
POWER BALANCE AND n0
• Assume low voltage sheaths at all surfacesET (Te) = Ec(Te)︸ ︷︷ ︸ + 2 Te︸︷︷︸ + 5.2 Te︸ ︷︷ ︸
Collisional Electron Ion• Power balance
Power in = power out
Pabs = (hln02πR2 + hRn02πRl) uB eET• Solve to obtain
n0 =Pabs
AeffuBeETwhere
Aeff = 2πR2hl + 2πRlhR
is an effective area for particle loss
• Density n0 is proportional to the absorbed power Pabs
• Density n0 depends on pressure p through hl, hR, and Te
LiebermanMinicourse10 33
University of California, Berkeley PLASMA
PARTICLE AND POWER BALANCE
• Particle balance =⇒ electron temperature Te
(independent of plasma density)
• Power balance =⇒ plasma density n0
(once electron temperature Te is known)
LiebermanMinicourse10 34
University of California, Berkeley PLASMA
EXAMPLE 1
• Let R = 0.15 m, l = 0.3 m, ng = 3.3× 1019 m−3 (p = 1 mTorrat 300 K), and Pabs = 800 W
• Assume low voltage sheaths at all surfaces
• Find λi = 0.03 m. Then hl ≈ hR ≈ 0.3 and deff ≈ 0.17 m[pp. 21, 27, 29]
• From the Te versus ngdeff figure, Te ≈ 3.5 V [p. 30]
• From the Ec versus Te figure, Ec ≈ 42 V [p. 23].Adding Ee = 2Te ≈ 7 V and Ei ≈ 5.2Te ≈ 18 V yieldsET = 67 V [p. 22]
• Find uB ≈ 2.9× 103 m/s and find Aeff ≈ 0.13 m2 [pp. 24, 33]
• Power balance yields n0 ≈ 2.0× 1017 m−3 [p. 33]
• Ion current density Jil = ehln0uB ≈ 2.9 mA/cm2
• Ion bombarding energy Ei ≈ 18 V [p. 31]
LiebermanMinicourse10 35
University of California, Berkeley PLASMA
EXAMPLE 2
• Apply a strong dc magnetic field along the cylinder axis=⇒ particle loss to radial wall is inhibited
• Assume no radial losses, then deff = l/2hl ≈ 0.5 m• From the Te versus ngdeff figure, Te ≈ 3.3 V (was 3.5 V)• From the Ec versus Te figure, Ec ≈ 46 V. Adding Ee = 2Te ≈ 6.6 V
and Ei ≈ 5.2Te ≈ 17 V yields ET = 70 V• Find uB ≈ 2.8× 103 m/s and find Aeff = 2πR2hl ≈ 0.043 m2
• Power balance yields n0 ≈ 5.8× 1017 m−3 (was 2× 1017 m−3)
• Ion current density Jil = ehln0uB ≈ 7.8 mA/cm2
• Ion bombarding energy Ei ≈ 17 V
=⇒ Slight decrease in electron temperature Te
=⇒ Significant increase in plasma density n0
EXPLAIN WHY!
• What happens to Te and n0 if there is a sheath voltageVs = 500 V at each end plate?
LiebermanMinicourse10 36
University of California, Berkeley PLASMA
ELECTRON HEATING MECHANISMS
• Discharges can be distinguished by electron heating mechanisms
(a) Ohmic (collisional) heating (capacitive, inductive discharges)
(b) Stochastic (collisionless) heating (capacitive, inductivedischarges)
(c) Resonant wave-particle interaction heating (Electron cyclotronresonance and helicon discharges)
• Although the heated electrons provide the ionization required tosustain the discharge, the electrons tend to short out the appliedheating fields within the bulk plasma
• Achieving adequate electron heating is a key issue
LiebermanMinicourse10 37
University of California, Berkeley PLASMA
INDUCTIVE DISCHARGES
DESCRIPTION AND MODEL
LiebermanMinicourse10 38
University of California, Berkeley PLASMA
MOTIVATION
• High density (compared to capacitive discharge)
• Independent control of plasma density and ion energy
• Simplicity of concept
• RF rather than microwave powered
• No source magnetic fields
LiebermanMinicourse10 39
University of California, Berkeley PLASMA
CYLINDRICAL AND PLANAR CONFIGURATIONS
• Cylindrical coil
• Planar coil
LiebermanMinicourse10 40
University of California, Berkeley PLASMA
HIGH DENSITY REGIME
• Inductive coil launches decaying wave into plasma
z
Coil
Plasma
E
H
Window
Decaying wave
δp
• Wave decays exponentially into plasma
E = E0 e−z/δp , δp =c
ω
1
Im(κ1/2p )
where κp = plasma dielectric constant [p. 16]
κp = 1− ω2pe
ω(ω − jνm)For typical high density, low pressure (νm ≪ ω) discharge
δp ≈c
ωpe∼ 1–2 cm
LiebermanMinicourse10 41
University of California, Berkeley PLASMA
TRANSFORMER MODEL
• For simplicity consider a long cylindrical discharge
Plasma
l
Rb
z
N turn coil Irf
Ipδp
• Current Irf in N turn coil induces current Ip in 1-turnplasma skin
=⇒ A transformer
LiebermanMinicourse10 42
University of California, Berkeley PLASMA
PLASMA RESISTANCE AND INDUCTANCE
• Plasma resistance Rp
Rp =1
σdc
circumference of plasma loop
average cross sectional area of loopwhere [p. 17]
σdc =e2nes
mνmwith nes = density at plasma-sheath edge
=⇒ Rp =πR
σdclδp
• Plasma inductance Lp
Lp =magnetic flux produced by plasma current
plasma current• Using magnetic flux = πR2µ0Ip/l
=⇒ Lp =µ0πR2
l
LiebermanMinicourse10 43
University of California, Berkeley PLASMA
COUPLING OF COIL TO PLASMA
• Model the sourceas a transformer
Vrf = jωL11Irf + jωL12Ip
Vp = jωL21Irf + jωL22Ip
• Transformer inductances
L11 =magnetic flux linking coil
coil current=
µ0πb2N 2
l
L12 = L21 =magnetic flux linking plasma
coil current=
µ0πR2Nl
L22 = Lp =µ0πR2
lLiebermanMinicourse10 44
University of California, Berkeley PLASMA
SOURCE CURRENT AND VOLTAGE
• Put Vp = −IpRp in transformer equations and solve for the
impedance Zs = Vrf/Irf seen at coil terminals
Zs = jωL11 +ω2L2
12
Rp + jωLp≡ Rs + jωLs
• Equivalent circuit at coil terminals
Rs = N 2 πR
σdclδp
Ls =µ0πR2N 2
l
(b2
R2− 1
)
• Power balance =⇒ Irf
Pabs =1
2I2rfRs
• From source impedance =⇒ Vrf
Vrf = IrfZs
LiebermanMinicourse10 45
University of California, Berkeley PLASMA
EXAMPLE
• Assume plasma radius R = 10 cm, coil radius b = 15 cm, length l = 20 cm,N = 3 turns, gas density ng = 6.6×1014 cm−3 (20 mTorr argon at 300 K),ω = 85 × 106 s−1 (13.56 MHz), absorbed power Pabs = 600 W, and lowvoltage sheaths
• At 20 mTorr, λi ≈ 0.15 cm, hl ≈ hR ≈ 0.1, deff ≈ 34 cm [pp. 21, 27, 29]• Particle balance (Te versus ngdeff figure [p. 30]) yields Te ≈ 2.1 V• Collisional energy losses (Ec versus Te figure [p. 23]) are Ec ≈ 110 V.
Adding Ee + Ei = 7.2Te yields total energy losses ET ≈ 126 V [p. 22]• uB ≈ 2.3 × 105 cm/s [p. 24] and Aeff ≈ 185 cm2 [p. 33]• Power balance yields ne ≈ 7.1 × 1011 cm−3 and nse ≈ 7.4 × 1010 cm−3
[p. 33]
• Use nse to find skin depth δp ≈ 2.0 cm [p. 41]; estimate νm = Kelng
(Kel versus Te figure [p. 20]) to find νm ≈ 3.4 × 107 s−1
• Use νm and nse to find σdc ≈ 61 Ω−1-m−1 [p. 17]• Evaluate impedance elements Rs ≈ 23.5 Ω and Ls ≈ 2.2 µH;
|Zs| ≈ ωLs ≈ 190 Ω [p. 45]• Power balance yields Irf ≈ 7.1A; from source impedance |Zs| = 190 Ω,
Vrf ≈ 1360 V [p. 45]LiebermanMinicourse10 46
University of California, Berkeley PLASMA
MATCHING DISCHARGE TO POWER SOURCE
• Consider an rf power source connected to a discharge
~
+
−
VT
ZT
I
V
+
−
ZL
Source Discharge
• Source impedance ZT = RT + jXT is givenDischarge impedance ZL = RL + jXL
• Time-average power delivered to discharge Pabs = 12Re (V I∗)
• For fixed source VT and ZT , maximize power delivered to discharge
XL = −XT
RL = RT
LiebermanMinicourse10 47
University of California, Berkeley PLASMA
MATCHING NETWORK
• Insert lossless matching network between power source and coil
Power source Matching network Discharge coil
• Continue EXAMPLE [p. 46] with Rs = 23.5 Ω and ωLs = 190 Ω;assume RT = 50 Ω (corresponds to a conductance 1/RT = 1/50 S)
• Choose C1 such that the conductance seen looking to the right atterminals AA′ is 1/50 S
=⇒ C1 = 71 pF
• Choose C2 to cancel the reactive part of the impedance seen at AA′
=⇒ C2 = 249 pF
• For Pabs = 600 W, the 50 Ω source supplies Irf = 4.9 A
• Voltage at source terminals (AA′) = IrfRT = 245 VLiebermanMinicourse10 48
University of California, Berkeley PLASMA
PLANAR COIL DISCHARGE
• Magnetic field produced by planar coil
• RF power is deposited in a ring-shaped plasma volume
Plasma
z
N turn coilIrf
Ip
δp
Primaryinductance
Couplinginductance
Plasmainductance
¾
• As for a cylindrical discharge, there is a primary (L11),coupling (L12 = L21) and secondary (Lp = L22) inductance
LiebermanMinicourse10 49
University of California, Berkeley PLASMA
PLANAR COIL FIELDS
• A ring-shaped plasma forms because
Induced electric field =
0, on axismax, at r ≈ 1
2Rwall
0, at r = Rwall
• Measured radial variation of Br (and Eθ) at three distances belowthe window (5 mTorr argon, 500 W, Hopwood et al, 1993)
LiebermanMinicourse10 50
University of California, Berkeley PLASMA
INDUCTIVE DISCHARGES
POWER BALANCE
LiebermanMinicourse10 51
University of California, Berkeley PLASMA
RESISTANCE AT HIGH AND LOW DENSITIES
• Plasma resistance seen by the coil [p. 45]
Rs = Rpω2L2
12
R2p + ω2L2
p
• High density (normal inductive operation) [p. 45]
Rs ∝ Rp ∝1
σdcδp∝ 1√
ne
• Low density (skin depth > plasma size)Rs ∝ number of electrons in the heating volume ∝ ne
Lowdensity
Highdensity
∝ ne
δp ~ plasma size
√ne∝ 1
ne
Rs
LiebermanMinicourse10 52
University of California, Berkeley PLASMA
POWER BALANCE
• Drive dischargewith rf current
• Power absorbed by discharge is Pabs = 12 |Irf |2Rs(ne) [p. 45]
Power lost by discharge Ploss ∝ ne [p. 33]
• Intersection (red dot) gives operating point; let I1 < I2 < I3
ne
Ploss
Pabs = 12 I2
1Rs
Pabs = 12 I2
2Rs
Pabs = 12 I2
3Rs
Power
• Inductive operation impossible for Irf ≤ I2
LiebermanMinicourse10 53
University of California, Berkeley PLASMA
CAPACITIVE COUPLING OF COIL TO PLASMA
• For Irf below the minimum current I2, there is only a weakcapacitive coupling of the coil to the plasma
Plasma
z
Ip
+
−Vrf
couplingCapacitive
• A small capacitive power is absorbed=⇒ low density capacitive discharge
ne
Ploss
Pabs = 12 I2
1Rs
Pabs = 12 I2
3Rs
Power
Cap
Ind
Cap Mode Ind Mode
LiebermanMinicourse10 54
University of California, Berkeley PLASMA
MEASURMENTS OF ARGON ION DENSITY
• Above 100 W, discharge is inductive and ne ∝ Pabs
• Below 100 W, a weak capacitive discharge is present
LiebermanMinicourse10 55
University of California, Berkeley PLASMA
REACTIVE NEUTRAL BALANCE IN DISCHARGES
LiebermanMinicourse10 56
University of California, Berkeley PLASMA
PLANE-PARALLEL DISCHARGE
• Example of N2 discharge with low fractional ionization (ng ≈ nN2)
and planar 1D geometry (l≪ R)
Pabs
A=πR2
N2 gas ninis
lnis
Low voltagesheaths
• Determine Te
Ion particle balance is [p. 29]
KizngnilA ≈ 2nisuBA
where nis = hlni with hl = 0.86/(3 + l/2λi)1/2 [p. 26]
Kiz(Te)
uB(Te)≈ 2hl
ngl=⇒ Te
LiebermanMinicourse10 57
University of California, Berkeley PLASMA
PLANE-PARALLEL DISCHARGE (CONT’D)
• Determine edge plasma density nis
Overall discharge power balance [p. 33] gives the plasma densityat the sheath edge
nis ≈Pabs
2eET uBA
• Determine central plasma density [p. 26]
ni =nis
hl
• Determine ion flux to the surface [p. 26]
Γis ≈ nisuB
• Determine ion bombarding energy [p. 31]
Ei = 5.2 Te
LiebermanMinicourse10 58
University of California, Berkeley PLASMA
REACTIVE NEUTRAL BALANCE
• For nitrogen atoms
e + N2Kdiss−→ 2N + e
• Assume low fractional dissociation and loss of N atoms onlydue to a vacuum pump Sp (m3/s)
AldnN
dt= Al 2Kdissngni − SpnNS = 0
• Solve for reactive neutral density at the surface
nNS = Kdiss2Alng
Spni
• Flux of N atoms to the surface
ΓNS =1
4nNS vN
where vN = (8kTN/πMN)1/2
LiebermanMinicourse10 59
University of California, Berkeley PLASMA
LOADING EFFECT
• Consider recombination and/or reaction of N atoms on surfaces
N + wallγrec−→ 1
2N2
N + substrateγreac−→ product
• Pumping speed Sp in the expression for nNS [p. 59] is replaced by
Sp −→ Sp + γrec1
4vN(2A−Asubs) + γreac
1
4vNAsubs
Asubs is the part of the substrate area reacting with N atoms
• nNS is reduced due to recombination and reaction losses
• nNS , and therefore etch and deposition rates, now depend onthe part of the substrate area Asubs exposed to the reactive neutrals,a loading effect
LiebermanMinicourse10 60
University of California, Berkeley PLASMA
ADSORPTION AND DESORPTION KINETICS
LiebermanMinicourse10 61
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ADSORPTION
• Reaction of a molecule with the surface
A + SKa−→←−Kd
A: S
• Physisorption (due to weak van der Waals forces)U
x
1–3 Å
0.01–0.25 V
• Chemisorption (due to formation of chemical bonds)U
x
1–1.5 Å
0.4–4 V
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University of California, Berkeley PLASMA
STICKING COEFFICIENT
• Adsorbed flux [p. 13]
Γads = sΓA = s · 14nAS vA
s(θ, T ) = sticking coefficientθ = fraction of surface sites covered with absorbatenAS = gas phase density of A near the surface
vA = (8kTA/πMA)1/2 = mean thermal speed of A
• Langmuir kineticss(θ, T) = s0(1− θ)
s0 = zero-coverage sticking coefficient (s0 ∼ 10−6–1)
s
s0
0 0 1 θ
Langmuirs0
0 0
1
T
LiebermanMinicourse10 63
University of California, Berkeley PLASMA
DESORPTION
A: SKd−→ A + S
• Rate coefficient has “Arrhenius” form
Kd = Kd0 e−Edesor/T
where Edesor = Echemi or Ephysi
• Pre-exponential factors are typically
Kd0 ∼ 1014–1016 s−1 physisorption
∼ 1013–1015 s−1 chemisorption
LiebermanMinicourse10 64
University of California, Berkeley PLASMA
ADSORPTION-DESORPTION KINETICS
• Consider the reactions
A + SKa−→←−Kd
A: S
• Adsorbed flux is [p. 63]
Γads = KanASn′
0(1− θ)
n′0 = area density (m−2) of adsorption sites
nAS = the gas phase density at the surface
Ka = s01
4vA/n′
0 [m3/s] (adsorption rate coef)
• Desorbed flux ∝ area density n′0θ of covered sites [p. 64]
Γdesor = Kdn′
0θ
LiebermanMinicourse10 65
University of California, Berkeley PLASMA
ADSORPTION-DESORPTION KINETICS (CONT’D)
• Equate adsorption and desorption fluxes (Γads = Γdesor)
=⇒ θ =KnAS
1 +KnAS
where K = Ka/Kd
00
1
5 10
θ
KnAS
“Langmuir isotherm” (T=const)
• Note that T ↑⇒ K ↓ ⇒ θ ↓
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University of California, Berkeley PLASMA
PLASMA-ASSISTED ETCH KINETICS
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University of California, Berkeley PLASMA
ION-ENHANCED PLASMA ETCHING
1. Low chemical etch rate of silicon substrate in XeF2 etchant gas
2. Tenfold increase in etch rate with XeF2 + 500 V argon ions,simulating ion-enhanced plasma etching
3. Very low “etch rate” due to the physical sputtering of silicon byion bombardment alone
LiebermanMinicourse10 68
University of California, Berkeley PLASMA
STANDARD MODEL OF ETCH KINETICS
• O atom etching of a carbon substrate
+
Ka Ki Kd Yi Ki
O CO
1 – θ θ
C(s) CO(s)
• Let n′0 = active surface sites/m2
• Let θ = fraction of surface sites covered with C : O bonds
O(g) + C(s)Ka−→ C : O (O atom adsorption)
C : OKd−→ CO(g) (CO thermal desorption)
ion + C : OYiKi−→ CO(g) (CO ion-assisted desorption)
LiebermanMinicourse10 69
University of California, Berkeley PLASMA
SURFACE COVERAGE
• The steady-state surface coverage is found from [pp. 65–66]dθ
dt= KanOS(1− θ)−Kdθ − YiKinisθ = 0
• nOS is the O-atom density near the surface
nis is the ion density at the plasma-sheath edge
• Ka is the rate coefficient for O-atom adsorption
Kd is the rate coefficient for thermal desorption of CO
Ki = uB/n′0 is the rate coefficient for ions incident on the surface
• Yi is the yield of CO molecules desorbed per ion incident on afully covered surface
Typically Yi ≫ 1 and Yi ≈ Yi0
√Ei − Ethr (as for sputtering)
=⇒ θ =KanOS
KanOS + Kd + YiKinis
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University of California, Berkeley PLASMA
ETCH RATES
• The flux of CO molecules leaving the surface isΓCO = (Kd + YiKinis) θ n′
0 [m−2-s−1]with n′
0 = number of surface sites/m2
• The vertical etch rate is
Ev =ΓCO
nC[m/s]
where nC is the carbon atom density of the substrate
• The vertical (ion-enhanced) etch rate is
Ev =n′
0
nC
1
1
Kd + YiKinis+
1
KanOS
• The horizontal (non ion-enhanced) etch rate is
Eh =n′
0
nC
1
1
Kd+
1
KanOS
LiebermanMinicourse10 71
University of California, Berkeley PLASMA
NORMALIZED ETCH RATES
EnC
Kdn0´
0
KanOS
Kd
0
2
4
6
8
10
4 8 12 16 20 24
Vertical (Ev)
Horizontal (Eh)
YiKinis = 5 Kd
• High O-atom flux ⇒ highest anisotropy Ev/Eh = 1 + YiKinis/Kd
• Low O-atom flux ⇒ low etch rates with Ev/Eh → 1
LiebermanMinicourse10 72
University of California, Berkeley PLASMA
SIMPLEST MODEL OF ION-ENHANCED ETCHING
• In the usual ion-enhanced regime YiKinis ≫ Kd
1
Ev= nC
1
Yi Kinisn′
0︸ ︷︷ ︸+
1
KanOSn′
0︸ ︷︷ ︸
Γis ΓOS
• The ion and neutral fluxes and the yield (a function of ion energy)determine the ion-assisted etch rate
• The discharge parameters set the ion and neutral fluxes and the ionbombarding energy
LiebermanMinicourse10 73
University of California, Berkeley PLASMA
ADDITIONAL CHEMISTRY AND PHYSICS
• Sputtering of carbonΓC = γsputKinisn
′
0
• Associative and normal desorption of O atoms,C : O −→ C + O(g)
2C : O −→ 2C + O2(g)
• Ion energy driven desorption of O atomsions + C : O −→ C + O(g)
• Formation and desorption of CO2 as an etch product
• Non-zero ion angular bombardment of sidewall surfaces
• Deposition kinetics (C-atoms, etc)
LiebermanMinicourse10 74
University of California, Berkeley PLASMA
SUMMARY
• Plasma discharges are widely used for materials processing and areindispensible for microelectronics fabrication
• The charged particle balance determines the electron temperatureand ion bombarding energy to the substrate =⇒ Yi(Ei)
• The energy balance determines the plasma density and the ion fluxto the substrate =⇒ Γis
• A transformer model determines the relation among voltage, cur-rent, and power for inductive discharges
• The reactive neutral balance determines the flux of reactive neutralsto the surface =⇒ ΓOS
• Hence the discharge parameters (power, pressure, geometry, etc) setthe ion and neutral fluxes and the ion bombarding energy
• The ion and neutral fluxes and the yield (a function of ion energy)determine the ion-assisted etch rate
LiebermanMinicourse10 75
University of California, Berkeley PLASMA
THANK YOUFOR ATTENDINGTHIS COURSE
MIKE LIEBERMAN
LiebermanMinicourse10 76
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